Bird dropping head a short distance (4)
I believe the answer is:
inch
'a short distance' is the definition.
(thesaurus)
'bird dropping head' is the wordplay.
'bird' becomes 'finch' (finch is a kind of bird).
'dropping head' means to remove the first letter.
'finch' with its first letter removed is 'INCH'.
(Other definitions for inch that I've seen before include "Move slowly and cautiously" , "Foot division" , "Crawl; imperial measurement" , "Move by slow degrees" , "Irish island" .)